Subspaces of $L^1$ spanned by the even and odd levels of the Haar system

In a recent paper, the discrete Gabor transform was connected to a Gabor transform with a time frequency domain given by the flat torus. We show that the corresponding Bargmann-Fock spaces can be expressed as theta functions (or equivalently line bundles on Abelian varieties). We give applications of this viewpoint to frame results for the discrete Gabor transform. In particular, we get necessary conditions which hold in higher dimensions and can expand the known results in the one dimensional case, the primary tool being the theorem of the square. We also give an application to asymptotics of restriction operators which arises via the asymptotic behavior of Bergman kernels and Toeplitz operators for high tensor powers of line bundles and find that time frequency restriction operators on the flat torus will exhibit "plunge" behaviors similar to those of time frequency restriction operators in other contexts.

math.FA↗

arXiv2026

On a minimal Andô dilation for a pair of strict contractions

The isometric dilation of a pair of commuting contractions due to Andô is not minimal. We modify Andô's dilation and construct a minimal isometric dilation on $\mathcal H \oplus_2 \ell_2(\mathcal H \oplus_2 \mathcal H)$ for a commuting pair of strict contractions on a Hilbert space $\mathcal H$. In the same spirit, we construct under certain conditions a minimal Andô dilation for a commuting pair of strict Banach space contractions. Further, we show that an Andô dilation is possible even for a more general pair of commuting contractions $(T_1,T_2)$ on a normed space $\mathbb X$ provided that the function $A_{T_i}: \mathbb X \rightarrow \mathbb R$ given by $A_{T_i}(x)=(\|x\|^2-\|T_ix\|^2)^{\frac{1}{2}}$ defines a norm on $\mathbb X$ for $i=1,2$.

math.FA↗

arXiv2026

Some properties of Fourier quasicrystals and measures on a strip

We extend certain results of the theory of Fourier quasicrystals on the real line to the case of a horizontal strip of finite width. For measures in a strip we use a natural generalization of the usual Fourier transform for measures on the line. We consider positive or translation bounded measures $μ$ on a strip whose Fourier transform is a pure point measure $\hatμ=\sum_{γ\inΓ}b_γδ_γ$ (as usual, $δ_γ$ is the unit mass at the point $γ$). We prove that the measure $ν=\sum_{γ\inΓ}|b_γ|^2δ_γ$ has the exponential growth. Moreover, if for some $η>0$ the points of $Γ$ in every interval of length $η$ are linearly independent over integers, then the measure $\hatμ$ also has the exponential growth.

math.FA↗